Primitive Complex Roots of Unity/Examples/Cube Roots

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Examples of Primitive Complex Roots of Unity

The primitive complex cube roots of unity are:

\(\ds \omega\) \(=\) \(\, \ds e^{2 \pi i / 3} \, \) \(\, \ds = \, \) \(\ds -\frac 1 2 + \frac {i \sqrt 3} 2\)
\(\ds \omega^2\) \(=\) \(\, \ds e^{4 \pi i / 3} \, \) \(\, \ds = \, \) \(\ds -\frac 1 2 - \frac {i \sqrt 3} 2\)


Proof

There are $3$ (complex) cube roots of unity:

$1, \omega, \omega^2$


We have:

\(\ds \omega\) \(=\) \(\ds e^{2 \pi i / 3}\)
\(\ds \omega^2\) \(=\) \(\ds e^{4 \pi i / 3}\)
\(\ds \omega^3\) \(=\) \(\ds 1\) Cube Roots of Unity


Also:

\(\ds \omega^2\) \(=\) \(\ds e^{4 \pi i / 3}\)
\(\ds \paren {\omega^2}^2\) \(=\) \(\ds \omega^3 \times \omega\)
\(\ds \) \(=\) \(\ds 1 \times \omega\) Cube Roots of Unity
\(\ds \) \(=\) \(\ds \omega\)
\(\ds \paren {\omega^2}^3\) \(=\) \(\ds \paren {\omega^3}^2\) Cube Roots of Unity
\(\ds \) \(=\) \(\ds 1 \times 1\) Cube Roots of Unity
\(\ds \) \(=\) \(\ds 1\)


Trivially, $1$ is not a primitive complex cube root of unity because you cannot make either $\omega$ or $\omega^2$ by multiplying $1$ by itself as many times as you like.

Hence the result.

$\blacksquare$